A consistent generalization of statistical mechanics is obtained by applying the maximum entropy principle to a trace-form entropy and by requiring that physically motivated mathematical properties are preserved. The emerging differential-functional equation yields a two-parameter class of generalized logarithms, from which entropies and power-law distributions follow: these distributions could be relevant in many anomalous systems. Within the specified range of parameters, these entropies possess positivity, continuity, symmetry, expansibility, decisivity, maximality, concavity, and are Lesche stable. The Boltzmann-Shannon entropy and some one parameter generalized entropies already known belong to this class. These entropies and their distribution functions are compared, and the corresponding deformed algebras are discussed.PACS numbers: 02.50.-r, 05.20.-y, 05.90.+m
By solving a differential-functional equation inposed by the MaxEnt principle
we obtain a class of two-parameter deformed logarithms and construct the
corresponding two-parameter generalized trace-form entropies. Generalized
distributions follow from these generalized entropies in the same fashion as
the Gaussian distribution follows from the Shannon entropy, which is a special
limiting case of the family. We determine the region of parameters where the
deformed logarithm conserves the most important properties of the logarithm,
and show that important existing generalizations of the entropy are included as
special cases in this two-parameter class.Comment: Presented at next2003, Second Sardinian International Conference on
News and Expectations in Thermostatistics, Villasimius (Cagliari) Italy,
21st-28th September 2003. In press Physica A (2004). Elsevier LaTeX macros,
11 pages, 1 figur
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